U.S. patent number 10,805,046 [Application Number 16/236,837] was granted by the patent office on 2020-10-13 for apparatus and method using polar code for multiple input multiple output (mimo) channel.
This patent grant is currently assigned to SAMSUNG ELECTRONICS CO., LTD.. The grantee listed for this patent is SAMSUNG ELECTRONICS CO., LTD.. Invention is credited to Prince Arya, Alok Jain, Kwang-chul Kim, Samir Kumar Mishra, Puneet Pandey.
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United States Patent |
10,805,046 |
Kim , et al. |
October 13, 2020 |
Apparatus and method using polar code for multiple input multiple
output (MIMO) channel
Abstract
A method of constructing a polar code for a multiple input
multiple output (MIMO) channel. The method may generate a plurality
of mutually independent single input single output (SISO) channels
based on MIMO channel information; define channels of the polar
code; allocate each of the channels of the polar code to one of the
plurality of the SISO channels; estimate qualities of channels
combined by applying polar transformation to the plurality of SISO
channels; and arrange unfrozen bits and frozen bits based on the
estimated qualities.
Inventors: |
Kim; Kwang-chul (Seongnam-si,
KR), Mishra; Samir Kumar (Karnataka, IN),
Jain; Alok (Karnataka, IN), Arya; Prince
(Karnataka, IN), Pandey; Puneet (Karnataka,
IN) |
Applicant: |
Name |
City |
State |
Country |
Type |
SAMSUNG ELECTRONICS CO., LTD. |
Suwon-si, Gyeonggi-Do |
N/A |
KR |
|
|
Assignee: |
SAMSUNG ELECTRONICS CO., LTD.
(Suwon-si, Gyeonggi-Do, KR)
|
Family
ID: |
1000005115236 |
Appl.
No.: |
16/236,837 |
Filed: |
December 31, 2018 |
Prior Publication Data
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Document
Identifier |
Publication Date |
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US 20190238268 A1 |
Aug 1, 2019 |
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Foreign Application Priority Data
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Jan 30, 2018 [KR] |
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10-2018-0011406 |
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Current U.S.
Class: |
1/1 |
Current CPC
Class: |
H04L
1/0058 (20130101); H03M 13/13 (20130101); H04B
7/0413 (20130101) |
Current International
Class: |
H03M
13/00 (20060101); H03M 13/13 (20060101); H04B
7/0413 (20170101); H04L 1/00 (20060101) |
References Cited
[Referenced By]
U.S. Patent Documents
Foreign Patent Documents
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10-1327505 |
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Nov 2013 |
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KR |
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1020170054046 |
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May 2017 |
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KR |
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Other References
Arikan, Erdal, "Channel polarization: A method for Constructing
Capacity-Achieving Codes for Symmetric Binary-Input Memoryless
Channels," IEEE Transactions on Information Theory 55, No. 7 (Jul.
2009), pp. 3051-3073. cited by applicant .
Wang X. et al., "On the Polar Codes for MIMO," IEEE International
Conference on Wireless Communications & Signal Processing
(WCSP), (2013), 5 pages. cited by applicant .
Mahdavifar, Hessam, et al., "Compound polar codes", Information
Theory and Application Workshop (ITA), 2013. IEEE, (2013), (6
pages). cited by applicant .
Kim J, Lee J., "Polar codes for non-identically distributed
channels", EURASIP Journal on Wireless Communications and
Networking. Dec. 1, 2016;2016(1):287 (17 pages). cited by
applicant.
|
Primary Examiner: Rizk; Samir W
Attorney, Agent or Firm: F. Chau & Associates, LLC
Claims
What is claimed is:
1. A method of constructing a polar code for a multiple input
multiple output (MIMO) channel, the method comprising: generating a
plurality of mutually independent single input single output (SISO)
channels based on MIMO channel information; executing, by at least
one processor, operations comprising: defining channels of the
polar code, and allocating each of the channels of the polar code
to one of the plurality of the SISO channels; estimating qualities
of channels combined by applying polar transformation to the
plurality of SISO channels; and arranging unfrozen bits and frozen
bits based on the estimated qualities.
2. The method of claim 1, wherein the generating of the plurality
of SISO channels comprises executing, by the at least one
processor, operations comprising: extracting a channel matrix
included in the MIMO channel information; and obtaining a diagonal
matrix having eigenvalues as diagonal elements from the channel
matrix based on singular value decomposition.
3. The method of claim 2, wherein the estimating of the qualities
of the combined channels comprises calculating a Bhattacharyya
parameter of each of the combined channels based on the
eigenvalues.
4. The method of claim 1, wherein the allocating comprises grouping
the plurality of SISO channels into at least one pair, and the
estimating of the qualities of the combined channels comprises
applying the polar transformation to the grouped plurality of SISO
channels.
5. The method of claim 4, wherein the grouping comprises grouping
based on an error rate of a block of a size corresponding to a
number of the SISO channels.
6. The method of claim 4, wherein the plurality of SISO channels is
r SISO channels, where r is an integer, and the plurality of SISO
channels respectively have first through r.sup.th indices of the
qualities in descending order, and the grouping, for a condition of
r being an even number, comprises grouping the plurality of SISO
channels into r/2 pairs such that a sum of indices of each pair is
(r+1).
7. The method of claim 1, wherein the arranging of the unfrozen
bits and the frozen bits comprises determining numbers of the
unfrozen bits and the frozen bits according to a code rate, and
sequentially arranging the unfrozen bits and the frozen bits in
descending order of the estimated qualities.
8. A communication method via a multiple input multiple output
(MIMO) channel, the communication method comprising: executing, by
at least one processor, operations comprising: obtaining MIMO
channel information; constructing a polar code based on the MIMO
channel information; and performing encoding or decoding according
to the polar code, wherein the constructing of the polar code
comprises allocating each of N channels of the polar code to one of
r layers of the MIMO channel, where N and r are positive
integers.
9. The communication method of claim 8, wherein the constructing of
the polar code further comprises generating r mutually independent
single input single output (SISO) channels based on a channel
matrix extracted from the MIMO channel information, and the
allocating comprises allocating each of a plurality of channels of
the polar code to one of the r SISO channels.
10. The communication method of claim 9, wherein the constructing
of the polar code further comprises: calculating Bhattacharyya
parameters of channels combined by applying polar transformation to
the r SISO channels; and arranging unfrozen bits and frozen bits
based on the Bhattacharyya parameters.
11. The communication method of claim 9, wherein the allocating
comprises grouping the r SISO channels into N/r pairs based on a
probability of block error in a block having a size N.
12. The communication method of claim 11, wherein the r SISO
channels respectively have first through r.sup.th indices in
descending order of signal to noise ratios thereof, and the
grouping, for a condition of r being an even number, comprises
grouping the r SISO channels into r/2 pairs such that a sum of
indices of each pair is (r+1).
13. The communication method of claim 10, wherein the arranging the
unfrozen bits and the frozen bits comprises: determining numbers of
the unfrozen bits and the frozen bits depending on a code rate; and
sequentially arranging the unfrozen bits and the frozen bits in
ascending order of the Bhattacharyya parameters.
14. The communication method of claim 9, further comprising
applying a precoding matrix generated from the channel matrix to
encoded data or applying a receiving matrix generated from the
channel matrix to data to be decoded.
15. The communication method of claim 8, wherein the MIMO channel
comprises a wireless communication channel generated via multiple
antennas and at least one modulator.
16. A communication device communicating via a multiple input
multiple output (MIMO) channel, the communication device
comprising: a polar code constructor circuit configured to
construct a polar code based on MIMO channel information; an
encoder circuit configured to generate N-bit code data by polar
coding K-bit source data based on the polar code, where K and N are
positive integers; and an interleaver circuit configured to
generate interleaved data comprising r groups corresponding to r
layers of the MIMO channel by interleaving the code data based on
the polar code, where r is a positive integer, wherein the polar
code constructor circuit comprises: a channel decomposer circuit
configured to generate r mutually independent single input single
output (SISO) channels based on a channel matrix extracted from the
MIMO channel; and a channel combiner circuit configured to generate
channels combined by applying polar transformation to the r SISO
channels.
17. The communication device of claim 16, wherein the channel
combiner circuit is configured to calculate Bhattacharyya
parameters of the combined channel and arrange unfrozen bits and
frozen bits based on the Bhattacharyya parameters.
18. The communication device of claim 16, wherein the r SISO
channels respectively have first through r.sup.th indices in
descending order of signal to noise ratios thereof, and the channel
combiner circuit, for a condition of r being an even number, is
configured to apply polar transformation to the r SISO channels
grouped into r/2 pairs such that a sum of indices of each pair is
(r+1).
19. The communication device of claim 16, further comprising: a
layer mapper circuit configured to distribute the interleaved data
to multiple output terminals of the MIMO channel; and a precoder
circuit configured to apply a precoding matrix to data distributed
to the multiple output terminals.
Description
CROSS-REFERENCE TO RELATED APPLICATION
This application claims the benefit of Korean Patent Application
No. 10-2018-0011406, filed on Jan. 30, 2018, in the Korean
Intellectual Property Office, the disclosure of which is
incorporated herein by reference in its entirety.
BACKGROUND
Technical Field
This disclosure relates generally to channel coding in data
communications over noisy channels to reduce communication errors.
More particularly, it relates to an apparatus and method utilizing
a polar code for a multiple input multiple output (MIMO)
channel.
Discussion of the Related Art
Channel coding may improve reliability of data transmission by
adding redundancy. A relatively new channel encoding method called
"polar coding" is a coding technique that forms polarized virtual
"bit-channels" using a class of block codes that contain polar
codes. The bit channels are categorized into "good channels" in
which data may be transmitted with high reliability, and "bad
channels" which are low reliability channels. A polar coding method
may substantially achieve Shannon capacity (the highest information
rate achievable with arbitrary small error probability) by using
low encoding/decoding complexity. In addition, MIMO may be used to
improve the capacity and the reliability of a channel. Accordingly,
in a system that integrates polar coding and MIMO, efficient
application of the polar code in a MIMO channel is desirable.
SUMMARY
Disclosed are apparatus and methods of constructing and using a
polar code for a multiple input multiple output (MIMO) channel.
According to an aspect of the inventive concept, there is provided
a method of constructing a polar code for a multiple input multiple
output (MIMO) channel. The method includes: generating a plurality
of mutually independent single input single output (SISO) channels
based on MIMO channel information; executing, by at least one
processor, operations comprising: defining channels of the polar
code and allocating each of the polar code channels to one of the
plurality of the SISO channels; estimating qualities of channels
combined by applying polar transformation to the plurality of SISO
channels; and arranging unfrozen bits and frozen bits based on the
estimated qualities.
According to another aspect of the inventive concept, there is
provided a communication method via a multiple input multiple
output (MIMO) channel, the communication method including
executing, by at least one processor, operations comprising:
obtaining MIMO channel information; constructing a polar code based
on the MIMO channel information; and performing encoding or
decoding according to the polar code, wherein the constructing of
the polar code, includes allocating each of N channels of the polar
code to one of r layers of the MIMO channel, where N and r are
positive integers.
According to another aspect of the inventive concept, there is
provided a communication device communicating via a multiple input
multiple output (MIMO) channel, the communication device including:
a polar code constructor circuit configured to construct a polar
code based on MIMO channel information; an encoder circuit
configured to generate, N-bit code data by polar coding K-bit
source data based on the polar code; and an interleaver circuit
configured to generate interleaved data comprising r groups
corresponding to r layers of the MIMO channel by interleaving the
code data based on the polar code, where K, N and r are each
positive integers.
A non-transitory computer-readable recording medium may store
instructions that, when executed by at least one processor,
implement the method of constructing the polar code or the
communication method summarized above.
BRIEF DESCRIPTION OF THE DRAWINGS
Embodiments of the inventive concept will be more clearly
understood from the following detailed description taken in
conjunction with the accompanying drawings in which like reference
characters denote like elements or functions, wherein:
FIG. 1 is a block diagram illustrating a communication system
according to an embodiment;
FIG. 2 is a block diagram illustrating an example of a polar code
constructor in FIG. 1 according to an embodiment;
FIG. 3 is a diagram illustrating a basic block of polar
transformation according to an embodiment;
FIG. 4 is a diagram illustrating an example of polar transformation
according to an embodiment;
FIG. 5 is a diagram illustrating an example of polar transformation
according to an embodiment;
FIG. 6 is a diagram illustrating an example of polar transformation
according to an embodiment;
FIG. 7 is a block diagram illustrating an example of a transmitter
in FIG. 1 according to an embodiment;
FIG. 8 is a flowchart of a communication method according to an
embodiment;
FIG. 9 is a flowchart of an example operation S400 in FIG. 8
according to an embodiment;
FIG. 10 is a flowchart of an example operation S400 in FIG. 8
according to an embodiment;
FIG. 11 is a flowchart of an example operation S480 in FIG. 9
according to an embodiment;
FIGS. 12A, 12B, and 12C are respective block diagrams illustrating
respective examples of systems according to embodiments; and
FIG. 13 is a block diagram of a communication device according to
embodiments.
DETAILED DESCRIPTION OF EMBODIMENTS
FIG. 1 is a block diagram illustrating a data communication system
10 including a transmitter 100 and a receiver 200 according to an
embodiment. Transmitter 100 and the receiver 200 may communicate
with each other via a multiple input multiple output (MIMO) channel
300. In the following description, variables representing a number
or indices are positive integers unless otherwise specified.
The system 10 may be any communication system that may benefit from
a MIMO based signal exchange over the MIMO channel 300. In some
embodiments, the system 10 may be, as a non-limiting example, a
wireless communication system such as a 5th generation (5G)
wireless system, a long term evolution (LTE) system, or a WiFi
system. In other embodiments, the system 10 may be a wired
communication system such as a storage system and a network system.
Hereinafter, the system 10 will be described mainly with reference
to the wireless communication system as an example.
Transmitter 100 may include a polar code constructor 110, an
encoder 130, an interleaver 150, a layer mapper 170, a precoder
190, and first through p.sup.th output terminals TX1 through TXp
that may respectively output transmitting signals {x.sub.1, . . . ,
x.sub.p}. Receiver 200 may receive q receiving signals {y.sub.1, .
. . , y.sub.q} via first through q input terminals RX1 through RXq.
When the system 10 is a wireless communication system, each of the
first through p.sup.th output terminals TX1 through TXp may be a
modulator and an antenna (or a single modulator may be used for all
the output terminals); and the first through q.sup.th input
terminals RX1 through RXq may each be a demodulator and an antenna
(or a single demodulator may be used for all the input terminals).
The transmitter 100 may include p output terminals TX through TXp
and the receiver 200 may include q input terminals RX1 through RXq.
The MIMO channel 300 may be represented as a channel matrix H
including h.sub.11 through h.sub.qp as elements, for a total of
(q.times.p) elements. A smaller number of the number p of the first
through p.sup.th output terminals TX1 through TXp and the number q
of the first through q input terminals RX1 through RXq may be
referred to as a rank of the MIMO channel 300. In other words, the
rank of the MIMO channel 300 may be defined as r=min (p, q), and
the MIMO channel 300 may be said to include r layers.
The polar code constructor 110 may construct the polar code for the
MIMO channel 300, where the polar code has a plurality of
"bit-channels". For example, the polar code constructor 110 may
obtain information about the MIMO channel 300 and may construct the
polar code based on such MIMO channel information. As described
below with reference to FIG. 2 and subsequent figures, the polar
code constructor 110 may divide the MIMO channel 300 into a
plurality of single input single output (SISO) channels, and may
allocate individual bit-channels of the polar code to respective
ones of the plurality of SISO channels. This approach may avoid the
need to use a plurality of encoders respectively corresponding to
the first through p.sup.th output terminals TX1 through TXp or a
plurality of decoders respectively corresponding to the first
through q.sup.th input terminals RX1 through RXq, for the purpose
of applying the polar code to the MIMO channel 300. Accordingly,
resources, for example, power, hardware resources, etc., which
would otherwise be used to apply the polar code to the MIMO channel
300 may be reduced, and the polar code to reduce a bit error rate
(BER) may be constructed (virtually) in the MIMO channel 300. The
polar code constructor 110 may control the encoder 130 and the
interleaver 150 according to the constructed polar code. Details of
an example polar code constructor 110 will be described below with
reference to FIG. 3 and other figures.
The encoder 130 may generate an N-bit code-word, that is, encoded
data D_ENC, by encoding K-bits source data D_SRC using the polar
code constructed by the polar code constructor 110. The polar code
may be based on channel polarization, which defines bit-channels
observed at an input stage as being polarized into good
bit-channels and bad bit-channels representing high reliability and
low reliability channels, respectively. (Hereafter, bit-channels of
the polar code may be interchangeably called sub-channels or just
channels of the polar code.) Accordingly, in the polar code,
information bits of the source data D_SRC, that is, unfrozen bits,
may be allocated to the good bit-channels while frozen bits having
known values at both sides of the transmitter 100 and the receiver
200 may be allocated to the bad bit-channels. Accordingly, the K
bits among the N bits may be "unfrozen" bits, while remaining bits
may be "frozen" bits. A code rate may be defined as R=K/N, where
the code rate R may be pre-defined and provided to the polar code
constructor 110.
The interleaver 150 may interleave the encoded data D_ENC based on
the polar code constructed by the polar code constructor 110 so as
to generate interleaved data D_INT including r groups respectively
corresponding to the r layers of the MIMO channel 300. As described
later with reference to FIG. 6, etc., the polar code constructor
110 may allocate each of the polar code channels to one of the SISO
channels based on the states of the SISO channels, and the
interleaver 150 may generate the interleaved data D_INT from the
encoded data D_ENC according to a result of the allocation of the
polar code constructor 110. Accordingly, as illustrated in FIG. 1,
the interleaved data D_INT including r groups, each of which has
N/r bits, may be generated. An example of the interleaver 150 will
be described below with reference to FIG. 7.
The layer mapper 170 may generate mapped data D_MAP to distribute
the interleaved data D_INT provided by the interleaver 150 to the
first through p.sup.th output terminals TX1 through TXp or to a
plurality of layers. (As noted earlier, each of the first through
p.sup.th output terminals TX1 through TXp may be a modulator and an
antenna, in which case any output terminal TXi may be considered at
least part of a "layer". Alternatively, modulators may be included
in the precoder 190 such that any output terminal TXi may be just
an antenna.) In some embodiments, the layer mapper 170 may evenly
distribute bits of the interleaved data D_INT. The precoder 190 may
apply a precoding matrix generated by singular value decomposition
(SVD) of the channel matrix H to the mapped data D_MAP provided by
the layer mapper 170 so that the transmitting signals {x.sub.1, . .
. , x.sub.p} are provided to the first through p.sup.th output
terminals TX1 through TXp, respectively.
The receiver 200 may perform operations corresponding to the
transmitter 100 so as to process the received signals {y.sub.1, . .
. , y.sub.q} received via the first through q.sup.th input
terminals RX1 through RXq. For example, a receiving matrix, which
is generated by performing the SVD of the channel matrix H, may be
applied to the receiving signals {y.sub.1, . . . , yq} received via
the input terminals RX1 to RXq. Next, a layer de-mapper (not shown)
of the receiver 200 may perform an inverted operation of the
operation performed by the layer mapper 170 of the transmitter 100
and a de-interleaver (not shown) may rearrange the bits of data
provided by the layer de-mapper by performing the inverted
operation of the operation performed by the interleaver 150 of the
transmitter 100. A decoder (not shown) may generate estimated
K-bits information bits by decoding the N-bit data provided by the
de-interleaver according to the polar code. In some embodiments,
the transmitter 100 in FIG. 1 may receive signals from the receiver
200 via the MIMO channel 300, and to this end, the transmitter 100
may include additional components to process the received signals.
Example embodiments of the present disclosure will be described
mainly with reference to the transmitter 100 below, but it should
be noted that the concepts of these embodiments are also similarly
applicable to the receiver 200.
FIG. 2 is a block diagram illustrating an example polar code
constructor 110 in FIG. 1 according to an embodiment. As described
above with reference to FIG. 1, polar code constructor 110 of FIG.
2 may construct the polar code for the MIMO channel 300 in FIG. 1,
and may control the encoder 130 and the interleaver 150 according
to the constructed polar code. As illustrated in FIG. 2, the polar
code constructor 110 may include a channel decomposer 112 and a
channel combiner 114.
The channel decomposer 112 may receive channel state information
CSI as channel information. The channel state information CSI may
be shared by the transmitter 100 and the receiver 200 in various
manners. For example, the receiver 200 may estimate a channel state
based on signals received from the transmitter 100, generate the
channel state information CSI from the estimated channel state, and
provide the generated channel state information CSI to the
transmitter 100. In some embodiments, the transmitter 100 and the
receiver 200 may share at least one index, where the index may
indicate one of a plurality of items included in a table shared by
the transmitter 100 and the receiver 200.
The channel decomposer 112 may generate a plurality of mutually
independent SISO channels from the MIMO channel 300 based on the
channel state information CSI.
In FIG. 1, the channel matrix H may be a q.times.p matrix and
represented as Formula 1 below through the SVD of the channel
matrix H. H=U.SIGMA.V.sup.H [Formula 1]
In Formula 1, V and U may form an orthonormal set, V may be
referred to as the precoding matrix, and U.sup.H may be referred to
as a receiving matrix. .SIGMA. may be a diagonal matrix having
eigenvalues, that is, {.lamda..sub.1, . . . , .lamda..sub.g} of the
channel matrix H, as diagonal elements. As described above with
reference to FIG. 1, when the precoding matrix V is applied to the
transmitter 100 and the receiving matrix U.sup.H is applied to the
receiver 200, the MIMO channel 300 may be represented by a new
channel matrix .SIGMA.. Since the new channel matrix .SIGMA. is a
diagonal matrix, there may be no interference between receiving
signals {y.sub.1, . . . , y.sub.q} in the receiver 200.
Accordingly, the MIMO channel 300 may be decomposed into r SISO
channels {W.sub.1, . . . , W.sub.r}. The receiving signals
{y.sub.1, . . . , y.sub.q} may have quality according to the
eigenvalues {.lamda..sub.1, . . . , .lamda..sub.r} of the new
channel matrix .SIGMA.. In other words, signal to noise ratios
(SNRs) of the respective SISO channels may be proportional to the
eigenvalues {.lamda..sub.1, . . . , .lamda..sub.r}. Since the
eigenvalues {.lamda..sub.1, . . . , .lamda..sub.r} are sorted in
descending order in the new channel matrix .SIGMA., that is,
(.lamda..sub.1.gtoreq..lamda..sub.2.gtoreq. . . .
.gtoreq..lamda..sub.r), the r SISO channels (W.sub.1, . . . ,
W.sub.r) may respectively have the qualities (e.g., the SNRs) in
descending order (SNR.sub.1.gtoreq.SNR.sub.2.gtoreq. . . .
.gtoreq.SNR.sub.r). As described later, information about the
quality of each SISO channel may be used to construct the polar
code.
The channel combiner 114 may receive the new channel matrix .SIGMA.
from the channel decomposer 112 and may receive the bit number N of
the codeword representing a magnitude of the polar code. The
channel combiner 114 may generate an N.times.N polar transformation
matrix from the bit number N of the codeword. The channel combiner
114 may generate a combined channel W.sub.C by combining r SISO
channels {W.sub.1, . . . , W.sub.r} which are represented by the
new channel matrix .SIGMA. provided from the channel decomposer
112. The combined channel W.sub.C may have indices of unfrozen bits
and frozen bits.
In some embodiments, the channel combiner 114 may estimate the
qualities of combined channels. For example, the channel combiner
114 may use the Bhattacharyya parameter as a value indicating the
quality of the SISO channels {W.sub.1, . . . , W.sub.r}. In a
binary input memoryless symmetric (BMS) channel, in which X and Y
indicate an input data set and an output data set, respectively,
having a bit transition probability W (y|x), the Bhattacharyya
parameter Z(W) may be defined as Formula 2 below.
Z(W)=.SIGMA..sub.y {square root over (W(y|0)W(y|1))}(x.di-elect
cons.X,y.di-elect cons.Y) [Formula 2]
In other words, the Bhattacharyya parameter may indicate a maximum
probability of a bit error occurring in the BMS channel. The
Bhattacharyya parameter in an additive white Gaussian noise (AWGN)
channel may be expressed as Formula 3 below when the SNR of the
channel is C.sub.SNR. Z(W)=exp(-C.sub.SNR) [Formula 3]
In other words, the Bhattacharyya parameter Z(W) may have a lower
value as the quality of the channel is higher (i.e., as the SNR is
higher).
When a length N of the codeword is 2.sup.n, that is, N=2.sup.n,
there may be n stages of polarization at a time of channel
combination. With j.di-elect cons.{0, 1, . . . , n-1} representing
the stages of polarization and Z.sub.j.sup.(i) representing the
Bhattacharyya parameter of an i.sup.th channel in a j.sup.th stage,
it may be possible that Z.sub.0.sup.(i)=Z(W), and the Bhattacharyya
parameter may be represented as Formula 4 below.
.times..ltoreq.<.ltoreq..ltoreq..times..times. ##EQU00001##
An example of an operation of the channel combiner 114 will be
described below with reference to FIGS. 3, 4, and 5.
FIG. 3 is a diagram illustrating a basic block of polar
transformation according to an embodiment. As shown in FIG. 3, the
basic block for polar transformation may be related to two BMS
channels {W.sub.1,W.sub.2}, and the two BMS channels
{W.sub.1,W.sub.2} may be transformed into binary input channels
{W.sub.C.sup.-,W.sub.C.sup.+} by applying the Arikan polarization
kernel F in Formula 5 below.
.times..times. ##EQU00002##
In the polar transformation in FIG. 3, the Bhattacharyya parameter
may be calculated as described in Formula 6 below.
Z(W.sub.C.sup.-).ltoreq.Z(W.sub.1)+Z(W.sub.2)-Z(W.sub.1)Z(W.sub.2)
Z(W.sub.C.sup.+)=Z(W.sub.1)Z(W.sub.2)
Z(W.sub.C.sup.+).ltoreq.Z(W.sub.C.sup.-) [Formula 6]
As explained further hereafter, the channels of the polar code may
be respectively allocated to the SISO channels, and when the
channels are combined, the Bhattacharyya parameters calculated from
the Bhattacharyya parameters of the SISO channels may be used in
the process of combining the channels.
In some embodiments, the r SISO channels may be grouped into r/2
pairs of SISO channels, and a combined channel W' may be generated
by combining the grouped SISO channels according to the polar
transformation. Accordingly, the channel W' may represent channels
combined from the r SISO channels at log.sub.2 r stages. Then, in
the same manner that the SISO channels have been combined,
identical copies of the channel W' may be combined at log.sub.2
N-log.sub.2 r stages. For example, when N is a multiple of r,
2.sup.(n-log.sup.2 .sup.r) copies of the channel W' may be
combined. Examples of channel combination will be described
below.
FIG. 4 is a diagram illustrating an example of polar transformation
according to an embodiment of the present disclosure. FIG. 4
illustrates an example of constructing a polar code of 4 bits in a
4.times.4 MIMO channel (that is, in this case, N=4 and n=2).
The MIMO channel may be decomposed into, for example, four mutually
independent SISO channels {W.sub.1,W.sub.2,W.sub.3,W.sub.4} by the
channel decomposer 112 in FIG. 2. Accordingly, the polar
transformation by Formula 6 may be performed at two stages
(log.sub.2 4=2). As illustrated in FIG. 4, at a first stage, two
pairs of the SISO channels {W.sub.1,W.sub.2} and {W.sub.3,W.sub.4}
may be transformed into two pairs of combined channels
{W.sub.A.sup.-,W.sub.A.sup.+} and {W.sub.B.sup.-,W.sub.B.sup.+}. At
a second stage, two pairs of channels {W.sub.A.sup.-,W.sub.A.sup.+}
and {W.sub.B.sup.-,W.sub.B.sup.+} may be transformed into two pairs
of combined channels {W.sub.AB.sup.--,W.sub.AB.sup.-+} and
{W.sub.AB.sup.+-,W.sub.AB.sup.++}. Finally, a combined channel W'
in which
W'={W.sub.AB.sup.--,W.sub.AB.sup.-+,W.sub.AB.sup.+-,W.sub.AB.sup.++}
may be generated. As described above with reference to FIG. 3,
2.sup.n-2 identical copies of the channel W' may be combined at
(n-2) stages. When the polar transformation is completed at n
stages, indices corresponding to the K lowest Bhattacharyya
parameters may be allocated to the unfrozen bits, while the
remaining (N-K) indices may be allocated to the frozen bits.
FIG. 5 is a diagram illustrating an example of polar transformation
according to an embodiment of the present disclosure. This example
illustrates constructing the polar code when the number of bits of
the codeword is 16 in a 4.times.4 MIMO channel (that is, r=4, N=16,
and n=4). In addition, FIG. 5 illustrates the Bhattacharyya
parameters calculated at each stage in a logarithmic domain, when
the SISO channels are sorted in descending orders of the channel
quality, and have SNRs of 12.5 dB, 5.8 dB, 2.4 dB, and -3.37 dB,
respectively (where a negative SNR signifies more noise power than
signal power).
The calculation for the Bhattacharyya parameters may be propagated
from the right side to the left side in FIG. 5. As illustrated, in
the first phase, the Bhattacharyya parameters may be calculated by
composing four SISO channels {W.sub.1,W.sub.2,W.sub.3,W.sub.4}.
Next, in the second phase, the Bhattacharyya parameters may be
calculated from four copies of the channel combined in the first
phase. Accordingly, the Bhattacharyya parameters may be calculated
for sixteen channels corresponding to the input bits {u.sub.1,
u.sub.2, . . . , u.sub.16} of the polar code.
The unfrozen bits and the frozen bits of the polar code may be
arranged based on the Bhattacharyya parameters. In other words,
each frozen bit may be allocated to a channel corresponding to a
Bhattacharyya parameter having a relatively higher value, and each
unfrozen bit may be allocated to a channel corresponding to a
Bhattacharyya parameter having a relatively lower value. For
example, when a code rate is about 0.5, in other words, when
unfrozen bits of eight bits and frozen bits of eight bits are
arranged, the eight frozen bits may be allocated to bits
{u.sub.1,u.sub.2,u.sub.3,u.sub.4,u.sub.5,u.sub.9,u.sub.10,u.sub.11}
corresponding to eight Bhattacharyya parameters having the greatest
values. Accordingly, the frozen bits may have indices
{1,2,3,4,5,9,10,11}. In addition, the unfrozen bits (or information
bits) of 8 bits may be allocated to bits
{u.sub.6,u.sub.7,u.sub.8,u.sub.12,u.sub.13,u.sub.14,u.sub.15,u.sub.16}
corresponding to eight Bhattacharyya parameters having the smallest
values. Thus, the unfrozen bits may have indices
{6,7,8,12,13,14,15,16}. In other words, in FIG. 1, the bits of the
K-bits source data D_SRC may be allocated to the bits corresponding
to K Bhattacharyya parameters having the lowest values among the N
bits, and the polar code constructor 110 may provide to the encoder
130 indices to which bits of the source data D_SRC are
allocated.
FIG. 6 is a diagram illustrating an example of polar transformation
according to an embodiment of the present disclosure. The example
illustrates a polar code of 4 bits in a 4.times.4 MIMO channel
(that is, N=4 and n=2) and unlike the example in FIG. 4, a channel
combined from two pairs of channels {W.sub.1,W.sub.4} and
{W.sub.2,W.sub.3} of the SISO channels
{W.sub.1,W.sub.2,W.sub.3,W.sub.4} may be generated.
As described above with reference to FIG. 5, in the process of
composing channels according to the polar transformation, the
Bhattacharyya parameter may depend on channels to be combined. For
example, in the first phase in FIG. 5, different values for the
Bhattacharyya parameters may result according to a combination of
the SISO channels {W.sub.1,W.sub.2,W.sub.3,W.sub.4}. In other
words, the Bhattacharyya parameters may be determined to differ
depending on a result of a grouping of the SISO channels
{W.sub.1,W.sub.2,W.sub.3,W.sub.4} into two pairs, that is, a
division of the SISO channels
{W.sub.1,W.sub.2,W.sub.3,W.sub.4}.
It may be possible to obtain r/2 pairs from r SISO channels
{W.sub.1, W.sub.2, . . . , W.sub.r}. With a parameter L defined as
a set having all possible combinations of pairs of the SISO
channels as elements, the number of elements of L may be calculated
as in Formula 7 below.
.times..times..times..times. ##EQU00003##
For example, when the 4.times.4 MIMO channel is decomposed into
four SISO channels {W.sub.1,W.sub.2,W.sub.3,W.sub.4}, L may have
three elements as shown in Formula 8 below.
L={{(W.sub.1,W.sub.2),(W.sub.3,W.sub.4)},{(W.sub.1,W.sub.3),(W.sub.2,W.su-
b.4)},{(W.sub.1,W.sub.4),(W.sub.2,W.sub.3)}} [Formula 8]
Similarly, in an 8.times.8 MIMO channel, it may be possible that
|L|=105, and in a 16.times.16 MIMO channel, it may be possible that
|L|=2027025. A parameter .pi..sub.opt, which represents an optimum
pairing of the SISO channels as elements of L, may minimize an
upper limit of a block error rate P.sub.b.sub.r for a block having
a size of r. .pi..sub.opt is represented as Formula 9 below.
.pi..times..times..di-elect cons..times..times..times.
##EQU00004##
Determining an optimum pairing for the SISO channels by calculating
the Bhattacharyya parameters in all cases based on Formula 9 may be
challenging when a change in the MIMO channel and real time
communication are considered. This issue may be resolved
deterministically as Formula 10 below.
.pi..times..times..times. ##EQU00005##
According to Formula 10, in the first stage of the polar code, a
SISO channel of the highest quality may be paired with a SISO
channel of the lowest quality, and a SISO channel of the second
highest quality may be paired with a SISO channel of the second
lowest quality. In other words, the r SISO channels may be grouped
into r/2 pairs such that a sum of the indices of each pair is
(r+1). For example, an optimum pairing of four SISO channels
{W.sub.1,W.sub.2,W.sub.3,W.sub.4} sorted in descending order may be
represented as Formula 11 below.
.pi..sub.opt={(W.sub.1,W.sub.4),(W.sub.2,W.sub.3)} [Formula 1]
Accordingly, as illustrated in FIG. 6, in the first stage, two
pairs {W.sub.1,W.sub.4} and {W.sub.2,W.sub.3} of the SISO channels
may be transformed into two pairs of combined channels
{W.sub.A.sup.-,W.sub.A.sup.+} and {W.sub.B.sup.-,W.sub.B.sup.+}. In
the second stage, two pairs {W.sub.A.sup.-,W.sub.A.sup.+} and
{W.sub.B.sup.-,W.sub.B.sup.+} of the SISO channels may be
transformed into two pairs of combined channels
{W.sub.AB.sup.--,W.sub.AB.sup.-+} and
{W.sub.AB.sup.+-,W.sub.AB.sup.++}. Finally, the combined channel
W'={W.sub.AB.sup.--,W.sub.AB.sup.-+,W.sub.AB.sup.+-,W.sub.AB.sup.++}
may be generated.
FIG. 7 is a block diagram illustrating a transmitter, 100', which
is an example of the transmitter 100 in FIG. 1 according to an
embodiment. Transmitter 100 includes four output terminals TX1
through TX4, polar code constructor 110', an encoder 130', and an
interleaver 150' that operates based on Formula 10, as well as
other components of transmitter 100 in FIG. 1 (e.g., layer mapper
170, precoder 190, not shown in FIG. 7). As described above with
reference to FIG. 1, the interleaver 150' in FIG. 7 may generate
interleaved data including four groups by interleaving encoded data
{x.sub.1, x.sub.2, . . . , x.sub.N} received from the encoder
130'.
As described above with reference to FIG. 6, the SISO channels may
be grouped into a plurality of pairs based on Formula 10, and the
polar transformation may be applied as a function of the pairs of
the grouped SISO channels. The interleaver 150' may be configured
such that the channel of the polar code corresponds to the SISO
channel according to Formula 10. For instance, as illustrated in
FIG. 7, the interleaver 150' may include N/4 (or first through
(N/4).sup.th) switch blocks 150_1 through 150_N/4, and each of the
first through (N/4) switch blocks 150_1 through 150_N/4 may receive
data of 4 bits among encoded data. Each of the first through
(N/4).sup.th switch blocks 150_1 through 150_N/4 may route bits of
the received data to one of the four layers, under the control of
the polar code constructor 110'. For example, the first switch
block 150_1 may receive {x.sub.1,x.sub.2,x.sub.3,x.sub.4}, and may
be configured as illustrated in FIG. 6 such that
{x.sub.1,x.sub.2,x.sub.3,x.sub.4} and
{W.sub.1,W.sub.2,W.sub.3,W.sub.4} are interrelated as per Formula
11.
FIG. 8 is a flowchart of a communication method according to an
embodiment. In particular, the communication method of FIG. 8 may
include operation S400 of constructing the polar code for the MIMO
channel according to an embodiment. For example, operations in FIG.
8 may be performed by the transmitter 100 and/or the receiver 200
in FIG. 1; thus the method of FIG. 8 will be described with
reference to FIG. 1.
MIMO channel information may be obtained (S200). For example, the
receiver 200 in FIG. 1 may estimate a state of the MIMO channel
based on a signal received from the transmitter 100. Transmitter
100 may also receive from the receiver 200 information about the
state of the MIMO channel 300 estimated by the receiver 200.
Accordingly, the transmitter 100 and the receiver 200 may share the
MIMO channel information. Note that the manner of acquiring the
MIMO channel information is not limited to the above-mentioned
examples.
The polar code for the MIMO channel may be constructed (S400). For
example, the MIMO channel may be decomposed into the plurality of
mutually independent SISO channels, and each of the channels of the
polar code may be allocated to one of the plurality of SISO
channels. (Examples of operation S400 will be described later with
reference to FIGS. 9 and 10.)
When data to be transmitted (Yes output of S610) is processed by
using the polar code generated in operation S400, encoding
operation S620 may be performed subsequently. The encoding may
involve receiving indices for the unfrozen bits and the frozen bits
provided by the polar code generated in operation S400. The bits of
data to be transmitted may be arranged in the unfrozen bits, and
the bits of predetermined fixed value may be arranged in the frozen
bits. Encoded data may be generated by applying the polar code to
the unfrozen bits and the frozen bits. Next, the precoding matrix
may be applied (S640). For example, the precoding matrix may be
obtained by performing the SVD on the channel matrix extracted from
the MIMO channel information obtained in operation S200, and the
precoding matrix may be applied to the encoded data.
When the received data is processed by using the polar code
constructed in operation S400 (NO output of S610), operation S820
of applying a receiving matrix may be performed subsequently. For
example, the receiving matrix may be obtained by performing the SVD
on the channel matrix extracted from the MIMO channel information
obtained in operation S200. The receiving matrix may be applied to
data received via the MIMO channel. Next, decoding may be performed
(S840). The frozen bits and the unfrozen bits of the decoded data
based on the indices for the unfrozen bits and the frozen bits
provided from the polar code constructed in operation S400 may be
distinguished.
FIG. 9 is a flowchart depicting an example of operation S400 in
FIG. 8 according to an embodiment. As described above with
reference to FIG. 8, constructing the polar code for the MIMO
channel may be performed in operation S400' of FIG. 9. As
illustrated in FIG. 9, operation S400' may include a plurality of
operations S420, S440, S460, and S480. For instance, operation
S400' may be performed by the polar code constructor 110 example of
FIG. 2.
The SISO channels may be generated (S420). For example, r SISO
channels may be generated by performing the SVD on the channel
matrix extracted from the MIMO channel information D81. An example
of operation S420 will be described below with reference to FIG.
10.
The channel of the polar code may be allocated to the SISO channel
(S440). For example, when the length of the codeword is N, each of
N channels of the polar code may be allocated to one of the r SISO
channels generated in operation S420. Accordingly, as illustrated
in the case that r=4 in the first phase in FIG. 5, positions of r
SISO channels may be determined, and the positions of r SISO
channels may be repeated. In addition, as described above with
reference to FIG. 6, r SISO channels may be paired to reduce the
bit error rate.
The polar transformation may be applied to the SISO channels
(S460). For example, as described above with reference to FIG. 5, a
channel combined from r SISO channels may be generated and copies
of the combined channels may be subsequently combined. In a
combining process of the channels, the qualities of the combined
channels may be estimated. An example of operation S460 will be
described below with reference to FIG. 10.
The unfrozen bits and the frozen bits may be arranged (S480). For
example, the unfrozen bits and the frozen bits may be arranged
according to the quality and the code rate of the channels
corresponding to each of the bits. Accordingly, the unfrozen bits
and the frozen bits may be allocated to the good channels and the
bad channels, respectively. An example of operation S480 will be
described below with reference to FIG. 11.
FIG. 10 is a flowchart of an example operation 8400 in FIG. 8
according to an embodiment. Specifically, operation S400'' of FIG.
10 may be an example of operations S420 and S460 of FIG. 9 and may
include operations S420' and S460'. The descriptions with respect
to FIG. 10 overlapping those with respect to FIG. 9 are omitted for
brevity.
The SISO channels may be generated (S420'), where operation S420'
may include operations S422 and S424. The channel matrix may be
extracted in operation S422 and the diagonal matrix may be obtained
in operation S424. When the rank of the MIMO channel is r, the
channel matrix may be an r.times.r matrix and the diagonal matrix
may be obtained by performing the SVD on the channel matrix.
Accordingly, r SISO channels may be obtained. The channel of the
polar code may be allocated to the SISO channel (S440').
The polar transformation may be applied to the SISO channels
(S460'). As illustrated in FIG. 10, operation S460' may include
operation S462 in which the Bhattacharyya parameter may be
calculated. As described above with reference to Formula 2, etc.,
the Bhattacharyya parameter may have a value indicating the quality
of the SISO channel as a maximum probability of occurrence of the
bit error. As illustrated in FIG. 5, in a process of the polar
transformation, the Bhattacharyya parameters of the combined
channels may be calculated and finally, the Bhattacharyya
parameters corresponding to each of the input bits of the polar
code may be calculated. Lastly, the unfrozen bits and the frozen
bits may be arranged (S480').
FIG. 11 is a flowchart of an example operation, S480', which is an
example of operation S480 in FIG. 9 according to an embodiment. As
described above with reference to FIG. 9, the operation of
arranging the unfrozen bits and the frozen bits may be performed in
operation S480. As illustrated in FIG. 11, operation S480' may
include operations S482 and S484.
The number of frozen bits may be determined (S482). For example, at
least two of the number N of bits, the code rate R, and the number
K of information bits of the code word of the polar code may be
obtained. Accordingly, the number of frozen bits (that is, N-K) and
the number of unfrozen bits (that is, K) may be determined.
The frozen bits may be arranged depending on the Bhattacharyya
parameters (S484). Bhattacharyya parameters of the channels
combined in the polar transformation process may be calculated so
that the frozen bits are allocated to the channels of low quality,
that is, of the highest value Bhattacharyya parameters.
Accordingly, the unfrozen bits, that is, information bits, may be
allocated to the channels of high quality, that is, of the lowest
value Bhattacharyya parameters.
FIGS. 12A, 12B, and 12C are block diagrams illustrating respective
examples of systems, 20, 30, and 40, according to embodiments. The
two entities included in each of the systems 20, 30, and 40 of
FIGS. 12A, 12B, and 12C may communicate unidirectionally and/or
bi-directionally via the MIMO channel, and the polar code may be
constructed and used for communication according to example
embodiments of the present disclosure described above.
Referring to FIG. 12A, the system 20 may be a wireless
communication system. A first wireless communication device 21 and
a second wireless communication device 22 may respectively include
a plurality of antennas, and may communicate with each other by
using a polar code for a MIMO channel 23. For example, the first
wireless communication device 21 may be a base station which is a
fixed station communicating with user equipment and/or another base
station. Examples of the first wireless communication device 21 may
include a Node B, an evolved Node B (eNB), a sector, a site, a base
transceiver system (BTS), an access point (AP), a relay node, a
remote radio head (RRH), a radio unit (RU), a small cell, etc. In
addition, the second wireless communication device 22 may be fixed
or mobile as user equipment, and may communicate with the base
station to transmit and receive data and/or control information.
Examples of the second wireless communication device 22 may include
terminal equipment, a mobile station (MS), a mobile terminal (MT),
a user terminal (UT), a subscriber station (SS), a handheld device,
and so forth.
Referring to FIG. 12B, the system 30 may be a storage system and
may include a host 31, a storage device 32, and an interface 33.
The host 31 may be a processor or a computing system including the
processor as a non-limiting example and may transmit a request of
write, read, erase, etc., to the storage device 32 via the
interface 33. The storage device 32 may include a semiconductor
memory device, as a non-limiting example, such as a flash memory, a
solid state drive (SSD), and a memory card, and may include a
storage medium different from the semiconductor memory device such
as a hard disk drive (HDD) and an optical disc. The storage device
32 may perform an operation corresponding to the request of the
host 31 and may provide a response to the host 31 via the interface
33.
The interface 33 may be modelled as the MIMO channel. For example,
a number of requests to write and read data may be transmitted in
parallel via the interface 33, and due to multiple requests having
such directional nature, the interface 33 may be modelled as the
MIMO channel and the polar code constructed according to
embodiments may be used.
Referring to FIG. 12C, the system 40 may be a network system, and
may include a terminal 41 and a server 42. The polar code
construction method according to the example embodiments may be
applied to an application layer forward error correction (AL-FEC).
For example, the system 40 may operate in accordance with a
multi-path transmission control protocol (TCP) (MPTCP) as a
multi-path transport protocol, and the terminal 41 and the server
42 may simultaneously transmit and receive data through a plurality
of TCP paths 43_1, 43_2, and 43_3, where the plurality of TCP paths
43_1, 43_2, and 43_4 may be modelled as the MIMO channel. The
AL-FEC may be modelled as an erasure channel when performance of
polar code construction is optimized.
The method of constructing the polar code according to embodiments,
as well as the examples illustrated in FIGS. 12A, 12B, and 12C, may
be used to improve performance of the bit-interleaved coded
modulation (BICM) method. The BICM method may provide unequal error
protection (UEP) to the bits mapped to modulation symbols and thus,
may be regarded as a virtual MIMO channel.
Apparatus and methods of constructing and using a polar code
according to the above-described embodiments may avoid using a
plurality of encoders respectively corresponding to the first
through p.sup.th output terminals TX1 through TXp or a plurality of
decoders respectively corresponding to the first through q.sup.th
input terminals RX1 through RXq, for the purpose of applying the
polar code to the MIMO channel 300 on the transmit side, and
decoding the polar code on the receive side. Accordingly, hardware
and processing resources that would otherwise be used to apply the
polar code to the MIMO channel 300, and accurately receive data
transmitted through the MIMO channel 300, may be reduced, and less
power may be consumed in the overall process as compared to
conventional art.
FIG. 13 is a block diagram of a communication device 50 according
to embodiments. As illustrated in FIG. 13, the communication device
50 may include an application specific integrated circuit (ASIC)
51, an application specific instruction set processor (ASIP) 53, a
memory 55, a main processor 57, and a main memory 59. Two or more
of the ASIC 51, the ASIP 53, and the main processor 57 may
communicate with each other. In addition, at least two of the ASIC
51, the ASIP 53, the memory 55, the main processor 57, and the main
memory 59 may be embedded in one chip.
The ASIP 53 may be an integrated circuit (IC) customized for a
particular application, support a dedicated instruction set for
particular applications, and execute instructions included in the
instruction set. The memory 55 may communicate with the ASIP 53 and
may store, as non-volatile storage, a plurality of instructions to
be executed by the ASIP 53. For example, the memory 55 may include,
as a non-limiting example, any type of memory accessible by the
ASIP 53 such as random access memory (RAM), read only memory (ROM),
a tape, a magnetic disk, an optical disc, a volatile memory, and a
combination thereof.
The main processor 57 may control the communication device 50 by
executing a plurality of instructions. For example, the main
processor 57 may control the ASIC 51 and the ASIP 53, and process
data received via the MIMO channel, or process user input to the
communication device 50. The main memory 59 may communicate with
the main processor 57 and may store a plurality of instructions
executed by the main processor 57 as a non-transitory storage
device. For example, the main memory 59 may include, as a
non-limiting example, any type of memory accessible by the ASIP 53
such as RAM, ROM, a tape, a magnetic disk, an optical disc, a
volatile memory, and a combination thereof.
The method(s) of constructing the polar code according to the
embodiment(s) described above may be performed by at least one or
substantially all of the components included in the communication
device 50 of FIG. 13. In some embodiments, at least one or
substantially all of the operations of the above-described method
of constructing the polar code may be implemented as a plurality of
instructions stored in the memory 55. In some embodiments, the ASIP
53 may perform at least one or substantially all of the operations
of the method of constructing the polar code by executing a
plurality of instructions stored in the memory 55. In some
embodiments, at least one or substantially all of the operations of
the method of constructing the polar code may be implemented in a
hardware block designed through logic synthesis or the like and be
included in the ASIC 51. In some embodiments, at least one or
substantially all of the operations of the method of constructing
polar code may be implemented as a plurality of instructions stored
in the main memory 59 and may be performed by the main processor 57
by executing the plurality of instructions stored in the main
memory 59.
Any one of the above-described components for manipulating,
generating and/or processing data and signals, such as any of the
above-described encoder, precoder, polar code constructor,
interleaver, layer mapper, precoder, channel decomposer, and
channel combiner, may be composed of electronic circuitry such as a
special purpose hardware circuit or processor or a general purpose
processor that executes instructions read from a memory to run a
routine to carry out the element's function. Various ones of the
above described components may be embodied as part of the same
processor, which executes instructions at different stages to carry
out the functions of the components sequentially, or using parallel
processing. With the use of parallel processing, various ones of
the components may be embodied as respective processing elements of
a parallel processor. Alternatively, the various components may be
embodied as part of a plurality of different processors. For
example, with such a composition based on hardware circuitry, the
above-discussed encoder, polar code constructor, interleaver, layer
mapper, precoder, channel decomposer, and channel combiner may
alternatively be called, respectively, an encoder circuit, polar
code constructor circuit, interleaver circuit, layer mapper
circuit, precoder circuit, channel decomposer circuit, and channel
combiner circuit, circuitry, processing element, processor,
computational hardware or the like.
As described above, embodiments have been disclosed in the drawings
and specification. While the embodiments have been described herein
with reference to specific terms, it should be understood that they
have been used only for the purpose of describing the technical
idea of the inventive concept and not for limiting the scope of the
inventive concept as defined in the claims. Further, while the
inventive concept has been particularly shown and described with
reference to embodiments thereof, it will be understood that
various changes in form and details may be made therein without
departing from the spirit and scope of the inventive concept as
defined by the following claims.
* * * * *